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I have a set of samples in MxN matrix X that I want to use to approximate a covariance matrix:

$$ C = X^\top X + eI $$

Where e is a small scalar and I is the NxN identity matrix.

I need draw a random sample from the normal distribution with covariance matrix $ C^{-1} $. I think this is normally done by drawing a random sample z from the standard normal, finding a factorization of $C^{-1} = M M^\top $ and then calculating $ y = M z$.

The difficulty is that N will in general be large so I need to avoid explicitly calculating C or other NxN matricies. This seems difficult to me, but I know that some L-BFGS methods do something a little like this so I have some hope. Any good ways of doing this?

One potentially convenient fact is that the $C^{-1} = {L^{-1}}^\top L^{-1}$ where $ C = L L^\top$ is the Cholesky decomposition of C. If I could solve $L^\top y = z$ for y without explicitly constructing L, that would solve my problem.

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1 Answer 1

Try the transformation $z=X^T a+\sqrt{\epsilon}b$ where a and b are standerd normal of dim M and N

You can calculate the covariant matrix and it is C.

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Yes, but the tricky part is that I need it with the covariance matrix C^-1 not C –  John Salvatier Jul 10 '12 at 15:00
    
Sorry for my mistake. –  Hoxide Jul 11 '12 at 1:31

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